Baseband equivalent volterra series for behavioral modeling and digital predistortion of wideband transmitters

ABSTRACT

A method, system and apparatus for modelling a power amplifier and pre-distorter fed by one of a wideband signal and an intra-band carrier aggregated signal are disclosed. According to one aspect, a method includes receiving the one of the wide-band signal and the intra-band carrier aggregated signal and generating a discrete base band equivalent, BBE, Volterra series based on the received signal, where the series has distortion products grouped according to determined shared kernels. The shared kernels are determined based on a transformation of a real-valued continuous-time pass band Volterra series without pruning of kernels.

TECHNICAL FIELD

The present invention relates to wireless communication systems, and inparticular to a method and system for use of Volterra series forbehavioral modeling and digital pre-distortion of wideband transmitters.

BACKGROUND

Development of energy efficient radio frequency (RF) power amplifiers(PAs) continues to be an area of focus for both industrial and academicresearchers. This interest has been motivated in part by the everincreasing high peak to average power ratio (PAPR) and bandwidth ofrecent communication signal standards. Innovative PA architectures foruse with extended back-off and bandwidth, such as Doherty PAs andenvelope tracking, have already been devised and successfullydemonstrated. However, the efficiency enhancement achieved by thesearchitectures has been compromised by the accentuation of signaldistortion problems. The search for a solution to address the distortionexhibited by these RF PAs has been a prolific area of research which hasled to the development of several linearization techniques; namelyfeed-forward, feedback and predistortion techniques.

While the predistortion technique, particularly its baseband digitalform, has been widely adopted to mitigate the distortions generated byRF PAs in 3^(rd) generation (3G) wireless base stations, currenttechniques do not meet the challenges of 4th generation (4G) wirelessnetworks using wideband and multi-standard signals. Investigators ofdigital predistortion (DPD) have struggled with linearization capacityand implementation complexity. Over time, DPD schemes have evolved fromsimple memoryless models to more sophisticated types which attempt tomitigate the memory effects that gain intensity as the signal bandwidthwidens.

Several DPD schemes with memory have been proposed, mainly consisting ofderivations of the low pass equivalent (LPE) Volterra series. In orderto address the complexity burden of the LPE Volterra series, thesederivations discard kernels that are considered negligible. One popularexample is the memory polynomial model, where several polynomials areapplied to the delayed input signals' samples separately and delayedcross-terms are discarded. Another approach, dynamic deviation reduction(DDR), has been employed to prune the LPE Volterra series. The approachbegins by reformulating an LPE Volterra series expression to assembleterms with the same dynamic distortion order. By setting a value for themaximum allowed dynamic order, such a reformulation allows a simpleelimination of distortion terms of higher dynamic order in the LPEVolterra series, consequently reducing its complexity. A DDR based LPEVolterra series with a preset value for the dynamic order equal to oneor two has been used to linearize various RF PAs and significantlinearization results were reported for different types of signals. Yet,as the signals' bandwidth increases, e.g., inter-band and inter-bandcarrier aggregated signals, the PA's memory effects gain intensity, andconsequently a larger value for the dynamic parameter of the DDRtechnique is needed and eventually the complexity of the pruned LPEVolterra series undesirably approaches the un-pruned series.

Recently, several multi-band behavioral models suitable for inter-bandcarrier aggregated scenarios have been reported. The models are suitedfor carrier aggregation scenarios when the frequency separation betweenthe two component carriers is significantly larger than the bandwidth ofthe individual component carriers. Consequently, these models operate ata speed proportional to the bandwidth of the individual componentcarriers and not to the frequency separation. However, in the case ofintra-band carrier aggregated signals, the frequency separation issignificantly lower. Hence, the spectrum regrowth (about five times thebandwidth of component carrier bandwidth) engendered at the differentcomponent carriers can overlap. This spectrum overlap hinders theapplicability of multi-band models since these models require theacquisition of the individual envelopes of the component carriers at theoutput of the PA. Hence, intra-band carrier aggregated systems need tobe handled as single-input single-output systems, and call for accurateand low-complex behavioral modelling schemes.

Some have distinguished two RF PA modeling strategies. The firstapproach approximates the behavior of the PA using its pass band inputand output signals. The inherent complexity associated with thisapproach has limited the pass band model's adoption in the area of RF PAmodeling and linearization. Instead, the LPE modeling approach is used.It is applied to the complex envelope of the RF input/output signals andrequires much simpler measurement hardware and reduced computation ascompared to its pass band counterpart. In fact, the LPE modelingapproach capitalizes on the band limited characteristics ofcommunication signals, thus limiting approximation efforts to the PAdistortions which affect the signal envelope around the carrierfrequency.

Essentially, the PA is treated as an envelope processing system.Therefore, if the real RF PA behavior is expressed as equation (1):

y(t)=ƒ(x(t)),  (1)

where x(t) is the real RF input signal, as shown by PA 2, y(t) is thereal RF output signal and ƒ is a describing function (modeling the RF PAbehavior), then the LPE methodology, as shown in FIG. 1, consists ofmodeling the equivalent envelope behavior of the PA via a low passtransformation, as shown by the PA 4. This is illustrated in equation(2)

{tilde over (y)}(n)={tilde over (ƒ)}({tilde over (x)}(n)),  (2)

where {tilde over (x)}(n) and y(n) denote the envelopes of the input andoutput signals, respectively, around the carrier and {tilde over (ƒ)}designates the LPE model.

Since a PA can be treated as a nonlinear dynamic system with fadingmemory, the Volterra series outlined in equation (3) is a suitablemodeling framework to approximate its behavior and/or synthesize thecorresponding predistortion module.

$\begin{matrix}{{y(t)} = {\sum\limits_{p = 1}^{NL}{\int_{- \infty}^{\infty}{\ldots \; {\int_{- \infty}^{\infty}{{h_{p}\left( {\tau_{1},\ldots \;,\tau_{p}} \right)}{\prod\limits_{j = 1}^{p}\; {{x\left( {t - \tau_{j}} \right)}{{\tau_{j}}.}}}}}}}}} & (3)\end{matrix}$

In this model, x(t) and y(t) designate the input and output RF pass bandsignals respectively, and h_(p) denotes the Volterra series' kernels.The direct application of the LPE strategy to the discrete input andoutput signals' envelopes yields the expression of equation (4) which iscommonly used for RF PA behavioral modeling.

$\begin{matrix}{{y(t)} = {\underset{p = {p + 2}}{\sum\limits_{p = 1}^{NL}}{\sum\limits_{i_{1} = 0}^{M}{\ldots \; {\sum\limits_{\frac{i_{p + 1}}{2} = \frac{i_{p - 1}}{2}}^{M}{\sum\limits_{\frac{i_{p + 3}}{2} = 0}^{M}{\ldots {\sum\limits_{i_{p} = i_{p - 1}}^{M}{{\overset{\sim}{h}}_{i_{1},\; \ldots \;,i_{p}} \cdot {\prod\limits_{j = 1}^{{({p + 1})}/2}\; {{\overset{\sim}{x}\left( {n - i_{j}} \right)} \cdot {\prod\limits_{j = \frac{({p + 3})}{2}}^{p}\; {{{\overset{\sim}{x}}^{*}\left( {n - i_{j}} \right)}.}}}}}}}}}}}}} & (4)\end{matrix}$

In equation (4) {tilde over (x)}(n) and {tilde over (y)}(n) designatethe input and output signals' envelopes around the carrier sampled atƒ_(s)=1/τ_(s), with n, NL, M and {tilde over (h)} representing thesample index the nonlinearity order, the memory depth and the LPEcomplex Volterra kernels respectively.

The LPE Volterra model of equation (4) has been used extensively in RFresearch. The model has been applied to develop solutions related tononlinear communication system modeling and estimation, satellitecommunication, digital transmission channel equalization, multichannelnonlinear CDMA system equalization, analysis and cancellation of theinter-carrier interference in nonlinear OFDM systems, decision feedbackequalization, nonlinear system and circuit analysis, data predistortion,PA modeling, and DPD. Although computationally more efficient than itspass band counterpart, the LPE Volterra series in its classical form (4)still suffers from a large number of kernels. As stated earlier, thisimpediment has been the key limitation to its widespread adoption for RFPA behavioral modeling and predistortion. Various attempts have beenmade to reduce the number of kernels required (e.g., Hammerstein,Weiner, Parallel Hammerstein, Parallel Weiner, DDR based Pruned Volterraseries) but at the cost of reduced modeling accuracy.

SUMMARY

Methods, systems and apparatus for modelling a power amplifier andpre-distorter fed by one of a wideband signal and an intra-band carrieraggregated signal are disclosed. According to one aspect, a methodincludes receiving the one of the wideband signal and the intra-bandcarrier aggregated signal and generating a discrete base bandequivalent, BBE, Volterra series based on the received signal, where theseries has distortion products grouped according to determined sharedkernels. The shared kernels are determined based on a transformation ofa real-valued continuous-time pass band Volterra series without pruningof kernels.

According to this aspect, in some embodiments, the shared kernels aredetermined based on the transformation of the real-valuedcontinuous-time pass band Volterra series by steps that includetransforming the real-valued continuous time pass band Volterra seriesto a multi-frequency complex-valued envelope series. The multi-frequencycomplex-valued envelope signal is transformed to a continuous-time passband-only series, which is then transformed to a continuous-timebaseband equivalent series. The continuous-time baseband equivalentsignal is discretized to produce the discrete base band equivalentVolterra series. Shared kernels are identified, each shared kernelhaving distortion products in common with another shared kernel. In someembodiments, transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal includes expressing thecontinuous-time pass band-only series in convolution form. The Laplacetransform is applied to the convolution form to produce a Laplace domainexpression, which is frequency shifted to baseband to produce a basebandequivalent expression in the Laplace domain. The inverse Laplacetransform is applied to the baseband equivalent expression to producethe continuous-time baseband equivalent series. In some embodiments, anumber of terms in the Laplace domain expression are reduced viasymmetry. In some embodiments, terms of the Laplace domain expressionare grouped based on frequency intervals where distortion terms are notzero. In some embodiments, discretizing the continuous-time basebandequivalent series to produce the discrete base band equivalent Volterraseries includes truncating the continuous-time baseband equivalentseries to a finite non-linearity order, and expressing the truncatedseries as summations of non-linear distortion terms, with upper limitsof the summations being memory depths, different memory depths beingassignable to different ones of the summations. In some embodiments, adistortion term is a group of distortion products multiplied by a sharedkernel.

According to another aspect, embodiments include a digital pre-distorter(DPD) system for pre-distortion of one of a wideband signal and anintra-band carrier aggregated signal. The system includes a Volterraseries DPD modelling unit. The DPD modelling unit is configured toreceive the one of the wideband signal and the intra-band carrieraggregated signal. The DPD modelling unit is also configured tocalculate a discrete base band equivalent, BBE, Volterra series based onthe received signal. The series has distortion products groupedaccording to determined shared kernels. The shared kernels aredetermined based on a transformation of a real-valued continuous-timepass band Volterra series without pruning of kernels.

According to this aspect, the DPD may further comprise a power amplifierconfigured to produce an output in response to the one of the widebandsignal and the intra-band carrier aggregated received signal. The outputof the power amplifier is provided to the Volterra series DPD modellingunit The Volterra series DPD modeling unit is configured to compute theshared kernels based on the output of the power amplifier. In someembodiments, the DPD system further comprises a transmitter observationreceiver configured to sample the output of the power amplifier andprovide the sampled output to the Volterra series DPD modelling unit. Insome embodiments, the distortion products and their associated kernelsare determined by transforming the real-valued continuous time pass bandVolterra series to a discrete base band equivalent Volterra seriesaccording to a series of steps that include: transforming thereal-valued continuous time pass band Volterra series to amulti-frequency complex-valued envelope series; transforming themulti-frequency complex-valued envelope signal to a continuous-time passband-only series; transforming the continuous-time pass band-only signalto a continuous-time baseband equivalent series; discretizing thecontinuous-time baseband equivalent signal to produce the discrete baseband equivalent Volterra series. The shared kernels are identified usingthe envelopes of the input and output signals of the PA. In someembodiments, transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal includes the following steps:expressing the continuous-time pass band-only series in convolutionform; applying a Laplace transform to the convolution form to produce aLaplace domain expression; frequency shifting the Laplace domainexpression to baseband to produce a baseband equivalent expression inthe Laplace domain; and applying an inverse Laplace transform to thebaseband equivalent expression to produce the continuous-time basebandequivalent series. In some embodiments, a number of terms in the Laplacedomain expression are reduced via symmetry. In some embodiments,transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal further includes groupingterms of the Laplace domain expression based on frequency intervalswhere distortion terms are not zero. In some embodiments, discretizingthe continuous-time baseband equivalent series to produce the discretebase band equivalent Volterra series includes: truncating thecontinuous-time baseband equivalent series to a finite non-linearityorder; and expressing the truncated series as summations of non-lineardistortion terms, with upper limits of the summations being memorydepths assigned to each order of the non-linear distortion terms. Insome embodiments, a distortion term is a group of distortion productsmultiplied by a shared kernel.

According to another aspect, embodiments include a Volterra seriesdigital pre-distorter, DPD, modelling unit. The DPD modelling unitincludes a memory module configured to store terms of a discrete baseband equivalent, BBE, Volterra series. The series is based on one of areceived wideband signal and an intra-band carrier aggregated signal.Also, a grouping module is configured to group distortion products ofthe series according to determined shared kernels. The DPD modellingunit also includes a shared kernel determiner configured to determinethe shared kernels based on a transformation of a real-valuedcontinuous-time pass band Volterra series without pruning of kernels.Also, a series term calculator is configured to calculate the terms ofthe discrete base band equivalent Volterra series, the terms being thedistortion products multiplied by their respective shared kernels.

In some embodiments, the shared kernel determiner is further configuredto determine the shared kernels via a least squares estimate based onthe one of the wideband signal and the intra-band carrier aggregatedreceived signal, and an output of a power amplifier. In someembodiments, the kernels and distortion products are derived from thereal-valued continuous-time pass band Volterra series by: transformingthe real-valued continuous time pass band Volterra series to amulti-frequency complex-valued envelope series; transforming themulti-frequency complex-valued envelope signal to a continuous-time passband-only series; transforming the continuous-time pass band-only signalto a continuous-time baseband equivalent series; discretizing thecontinuous-time baseband equivalent signal to produce the discrete baseband equivalent Volterra series; and identifying the shared kernels,each shared kernel having distortion products in common. In someembodiments, transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal includes: expressing thecontinuous-time pass band-only series in convolution form; applying aLaplace transform to the convolution form to produce a Laplace domainexpression; frequency shifting the Laplace domain expression to basebandto produce a baseband equivalent expression in the Laplace domain; andapplying an inverse Laplace transform to the baseband equivalentexpression to produce the continuous-time baseband equivalent series.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the present invention, and theattendant advantages and features thereof, will be more readilyunderstood by reference to the following detailed description whenconsidered in conjunction with the accompanying drawings wherein:

FIG. 1 is a diagram of low pass transformation for a low pass equivalent(LPE) power amplifier;

FIG. 2 is a column chart showing numbers of kernels for differentexemplary Volterra series implementations;

FIG. 3 is a spectrum of a PA output for several different linearizationmethods for a wideband input signal;

FIG. 4 is a spectrum a PA output for several different linearizationmethods for an intra-band carrier aggregated input signal, for a firstPA;

FIG. 5 is a spectrum a PA output for several different linearizationmethods for an intra-band carrier aggregated input signal, for a secondPA;

FIG. 6 is a block diagram of an exemplary power amplification systemhaving a digital pre-distorter modelling unit implementing the BBEVolterra model presented herein;

FIG. 7 is a block diagram of the pre-distorter modelling unit of FIG. 6;

FIG. 8 is a flowchart of an exemplary process for modelling a poweramplifier fed by a wideband or intra-band carrier aggregated inputsignal;

FIG. 9 is a flowchart of an exemplary process for transforming areal-valued continuous time pass band Volterra series to a discrete baseband equivalent Volterra series in which shared kernels are identified;

FIG. 10 is a flowchart of an exemplary process of transforming thecontinuous-time pass band-only signal to a continuous-time basebandequivalent signal; and

FIG. 11 is a flowchart of an exemplary process of discretizing thecontinuous-time baseband equivalent series to produce the discrete baseband equivalent Volterra series.

DETAILED DESCRIPTION

Before describing in detail exemplary embodiments that are in accordancewith the present invention, it is noted that the embodiments resideprimarily in combinations of apparatus components and processing stepsrelated to use of baseband equivalent Volterra series for digitalpre-distortion of power amplifiers driven by wideband or intra-bandcarrier aggregated signals. Accordingly, the system and methodcomponents have been represented where appropriate by conventionalsymbols in the drawings, showing only those specific details that arepertinent to understanding the embodiments of the present invention soas not to obscure the disclosure with details that will be readilyapparent to those of ordinary skill in the art having the benefit of thedescription herein.

As used herein, relational terms, such as “first” and “second,” “top”and “bottom,” and the like, may be used solely to distinguish one entityor element from another entity or element without necessarily requiringor implying any physical or logical relationship or order between suchentities or elements.

A base band equivalent (BBE) expression of the Volterra series forlinearization of a power amplifier driven by one of a wideband signaland an intra-band carrier aggregated signal is provided and theprocedure to derive this new expression is described below in detail. Awideband signal and an intra-band carrier aggregated signal aredistinguishable from a multi-band signal. A multi-band signal hascomponent carriers in different communication bands. Communication bandsmay be defined by the third generation partnership project (3GPP)standards. In contrast to a multi-band signal, a wideband signal hascomponent carriers that are contiguous in the same communication band.An intra-band carrier aggregated signal has carriers that may becontiguous or non-contiguous, but are in the same communication band.

The complexity and performance of the compact BBE Volterra seriesdisclosed herein is compared to those of the widely used LPE Volterraseries expression. For purposes of clarity, a simplified version of theLPE Volterra series given by equation (4), with nonlinearity order NL=3and memory effects order M=1 (i.e. {tilde over (x)}(t−T_(s))), will beused henceforward as per equation (5).

$\begin{matrix}\begin{matrix}{{\overset{\sim}{y}(n)} = {{\sum\limits_{i_{1} = 0}^{1}{{\overset{\sim}{h}}_{i_{1}}{{\overset{\sim}{x}}_{1}\left( {n - i_{1}} \right)}}} + {\sum\limits_{i_{1} = 0}^{1}{\sum\limits_{i_{2} = i_{1}}^{1}{\sum\limits_{i_{3} = 0}^{1}{{\overset{\sim}{h}}_{{i_{1}i_{2}},i_{3}} \cdot}}}}}} \\{{{\overset{\sim}{x}\left( {n - i_{1}} \right)}{\overset{\sim}{x}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{3}} \right)}}} \\{= {{{\overset{\sim}{h}}_{0}{\overset{\sim}{x}(n)}} + {{\overset{\sim}{h}}_{1}{\overset{\sim}{x}\left( {n - 1} \right)}} + {{\overset{\sim}{h}}_{0,0,0}{\overset{\sim}{x}(n)}{\overset{\sim}{x}(n)}{\overset{\sim}{x}}^{*}(n)} +}} \\{{{{\overset{\sim}{h}}_{0,0,1}{\overset{\sim}{x}(n)}{\overset{\sim}{x}(n)}{{\overset{\sim}{x}}^{*}\left( {n - 1} \right)}} +}} \\{{{{\overset{\sim}{h}}_{0,1,0}{\overset{\sim}{x}(n)}{\overset{\sim}{x}\left( {n - 1} \right)}{{\overset{\sim}{x}}^{*}(n)}} +}} \\{{{{\overset{\sim}{h}}_{0,1,1}{\overset{\sim}{x}(n)}{\overset{\sim}{x}\left( {n - 1} \right)}{{\overset{\sim}{x}}^{*}\left( {n - 1} \right)}} +}} \\{{{{\overset{\sim}{h}}_{1,1,0}{\overset{\sim}{x}\left( {n - 1} \right)}{\overset{\sim}{x}\left( {n - 1} \right)}{{\overset{\sim}{x}}^{*}(n)}} +}} \\{{{\overset{\sim}{h}}_{1,1,1}{\overset{\sim}{x}\left( {n - 1} \right)}{\overset{\sim}{x}\left( {n - 1} \right)}{{{\overset{\sim}{x}}^{*}\left( {n - 1} \right)}.}}}\end{matrix} & (5)\end{matrix}$

The implementation of the BBE Volterra series, suitable for thebehavioral modeling and predistortion of RF PAs, uses a number of stepsthat are described in detail as follows:

Step 1: Continuous-Time Real-Valued Volterra Series Modeling.

The Volterra series framework is initially used to describe therelationship between the real passband signals located at the systeminput and output stages as follows:

$\begin{matrix}{{y(t)} = {\sum\limits_{p = 1}^{NL}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{h_{p}\left( {\tau_{1},\ldots \;,\tau_{p}} \right)}{\prod\limits_{j = 1}^{p}\; {{x\left( {t - \tau_{j}} \right)}{\tau_{j}}}}}}}}} & (6)\end{matrix}$

where x(t) and y(t) represent the PA input and output RF signals and NLis the nonlinearity order.

Step 2: Real-Valued to Complex-Valued Envelope Signal Transformation.The input signal x(t) is a band limited modulated signal that can beexpressed as a function of its envelope around the carrier angularfrequency co, as follows:

$\begin{matrix}\begin{matrix}{{x(t)} = {{Re}\left\{ {{\overset{\sim}{x}(t)}^{j\; \omega_{c}t}} \right\}}} \\{{= {\frac{1}{2}\left( {{{{\overset{\sim}{x}}^{*}(t)}^{{- {j\omega}_{c}}t}} + {{\overset{\sim}{x}(t)}^{j\; \omega_{c}t}}} \right)}},}\end{matrix} & (7)\end{matrix}$

In the equation above, {tilde over (x)}(t) represents the complexbaseband envelope signal that modulates the angular frequency ω_(c).Substituting equation (7) into equation (6) yields an expressionrelating the output signal y(t) to {tilde over (x)}(t) and ω_(c) asfollows:

y(t)=ƒ({tilde over (x)}(t),e ^(±jpω) ^(c) ^(t));pε{1, . . . ,NL},  (8)

where the describing function ƒ is used to represent the real valuedVolterra series of equation (6). Since the output signal y(t) inequation (8) has been altered by the application of a nonlinear functionto a band-limited RF signal, it now contains several spectrum componentsthat involve multiple envelopes, represented by {tilde over (y)}_(p)(t).The spectrum components modulate the mixing products of ω_(c) asfollows:

$\begin{matrix}\begin{matrix}{{y(t)} = {\sum\limits_{p = {- {NL}}}^{p = {NL}}{\frac{1}{2}\left( {{{\overset{\sim}{y}}_{p}*{(t) \cdot ^{{- j}\; p\; \omega_{c}t}}} + {{{\overset{\sim}{y}}_{p}(t)} \cdot ^{j\; p\; \omega_{c}t}}} \right)}}} \\{= {{\frac{1}{2}\left( {{{\overset{\sim}{y}}_{0}*{(t) \cdot ^{{- 0}\; j\; t}}} + {{{\overset{\sim}{y}}_{0}(t)} \cdot ^{0\; j\; t}}} \right)} +}} \\{{{\frac{1}{2}\left( {{{\overset{\sim}{y}}_{1}*{(t) \cdot ^{{- j}\; \omega_{c}t}}} + {{{\overset{\sim}{y}}_{1}(t)}^{j\; \omega_{c}t}}} \right)} +}} \\{{{\frac{1}{2}\left( {{{\overset{\sim}{y}}_{2}*{(t) \cdot ^{{- j}\; 2\; \omega_{c}t}}} + {{{\overset{\sim}{y}}_{2}(t)}^{j\; 2\; \omega_{c}t}}} \right)} +}} \\{\vdots} \\{{{\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{NL}^{*}(t)} \cdot ^{{- {NL}}\; j\; \omega_{c}t}} + {{{\overset{\sim}{y}}_{NL}(t)}^{{NL}\; j\; \omega_{c}t}}} \right)},}}\end{matrix} & (9)\end{matrix}$

where {tilde over (y)}₀(t) denotes the envelope at DC, {tilde over(y)}₁(t) denotes the envelopes of the first harmonic in-band signals,{tilde over (y)}₂ (t) denotes the envelopes of the second order harmonicsignals and {tilde over (y)}_(NL) (t) represents the NL^(th) harmonicsignal.

Step 3: Multi-Frequency to Passband Only Transformation.

By equating the terms that share the same frequency range (fundamental,mixing product) from the right sides of equations (8) and (9), amulti-harmonic model is derived which consists of several distinctequations relating the output envelopes {tilde over (y)}_(p)(t) to{tilde over (x)}(t) and ω_(c). Since interest is in the relationshipbetween the envelopes of the output and input signals around the carrierfrequency, only the pass band component of the PA output is consideredin the equation below:

$\begin{matrix}\begin{matrix}{{y_{pb}(t)} = {\frac{1}{2}\left( {{{{\overset{\sim}{y}}_{1}^{*}(t)} \cdot ^{{- j}\; \omega_{c}t}} + {{{\overset{\sim}{y}}_{1}(t)}^{j\; \omega_{c}t}}} \right)}} \\{{= {\frac{1}{2}\left( {{y_{1}^{*}(t)} + {y_{1}(t)}} \right)}},}\end{matrix} & (10)\end{matrix}$

where y₁(t)={tilde over (y)}₁(t)e^(jω) ^(c) ^(t). The output signalenvelope y_(pb)(t) in equation (10) has the form of a band limitedsignal as given in equation (7).

Step 4: Continuous Time Passband Volterra Series.

The following derivation is presented for the term around e^(jω) ^(c)^(t) only; however, the same equation can be used to derive theconjugate term. The PA passband signal y₁(t) can be modeled as asummation of the Volterra series nonlinear order responsesy_(1,2k+1)(t), or as the nonlinear order envelope responses {tilde over(y)}_(1,2k+1)(t), around the angular frequency ω_(c) as follows:

$\begin{matrix}{{y_{1}(t)} = {{\sum\limits_{k = 0}^{\infty}{y_{1,{{2\; k} + 1}}(t)}} = {\left( {\sum\limits_{k = 0}^{\infty}{{\overset{\sim}{y}}_{1,{{2\; k} + 1}}(t)}} \right) \cdot ^{j\; \omega_{c}t}}}} & (11)\end{matrix}$

Note that only odd powered terms are retained; even terms are discardedsince they do not appear around the carrier frequency. Equating theterms on the right sides of the expanded equations (8) and (10) yield acontinuous BBE Volterra series that expresses y_(1,2k+1)(t) as afunction of {tilde over (x)}(t) and ω_(c). Below is the expressionderived using y_(1,1)(t) and y_(1,3)(t):

$\begin{matrix}{\mspace{79mu} {{y_{1,1}(t)} = {\int_{- \infty}^{\infty}{{{h_{1}\left( \tau_{1} \right)} \cdot {\overset{\sim}{x}\left( {t - \tau_{1}} \right)}}{^{j\; {\omega_{c}{({t - \tau_{1}})}}} \cdot {\tau_{1}}}}}}} & (12) \\{{y_{1,3}(t)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{h_{3}\left( {\tau_{1},\tau_{2},\tau_{3}} \right)} \cdot \left\{ {{\left( {{{\overset{\sim}{x}\left( {t - \tau_{1}} \right)}{^{j\; {\omega_{c}{({t - \tau_{1}})}}}\left( {{\overset{\sim}{x}\left( {t - \tau_{2}} \right)}^{j\; {\omega_{c}{({t - \tau_{2}})}}}} \right)}\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)^{j\; {\omega_{c}{({t - \tau_{3}})}}}} \right)^{*}} + {\left( {{\overset{\sim}{x}\left( {t - \tau_{1}} \right)}^{j\; {\omega_{c}{({t - \tau_{1}})}}}} \right)\left( {{\overset{\sim}{x}\left( {t - \tau_{2}} \right)}^{j\; {\omega_{c}{({t - \tau_{2}})}}}} \right)^{*}\left( {{\overset{\sim}{x}\left( {t - \tau_{3}} \right)}^{j\; {\omega_{c}{({t - \tau_{3}})}}}} \right)} + {\left( {{\overset{\sim}{x}\left( {t - \tau_{1}} \right)}^{j\; {\omega_{c}{({t - \tau_{1}})}}}} \right)^{*}\left( {{\overset{\sim}{x}\left( {t - \tau_{2}} \right)}^{j\; {\omega_{c}{({t - \tau_{2}})}}}} \right)\left( {{\overset{\sim}{x}\left( {t - \tau_{3}} \right)}^{j\; {\omega_{c}{({t - \tau_{3}})}}}} \right)}} \right\} \cdot {\tau_{3}}}{\tau_{2}}{\tau_{1}}} \right.}}}}} & (13)\end{matrix}$

Step 5: Continuous-Time Passband to Baseband Equivalent Transformation.

In order to be computed by a digital processor, with manageablecomplexity, a passband model should be transformed to a BBE model. A BBEmodel includes the RF nonlinear dynamic distortion and computes theterms in baseband with a low sampling rate. To translate the passbandmodel to baseband, the continuous time passband Volterra series ofequations (10) and (12) is first rewritten using convolution form. Theconvolution form for y_(1,1)(t) is given by:

y _(1,1)(t)=h ₁(t)*(x(t)e ^(jω) ^(c) ^(t)).  (14)

As the kernel h₃ is tri variate, h₃(τ₁,τ₂,τ₃), and the output y(t) ismono-variate, the output function is re-assigned as follows:

y(t)=y(t ₁ ,t ₂ ,t ₃)_(|t) ₁ _(=t) ₂ _(=t) ₃ _(=t)

y(t,t,t).

The convolution form is given as:

$\begin{matrix}{{y_{1,3}\left( {t_{1},t_{2},t_{3}} \right)} = {{h_{3}\left( {t_{1},t_{2},t_{3}} \right)}*{\left\{ {{\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)^{*}} + {\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)^{*}\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)} + {\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)^{*}\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)}} \right\}.}}} & (15)\end{matrix}$

Next, the Laplace transformation is applied to the convolution form ofthe Volterra series nonlinear order responses y_(1,2k+1). The Laplacedomain representation of equations (14) and (15) is given by:

$\begin{matrix}{\mspace{79mu} \begin{matrix}{{Y_{1,1}(s)} = {\mathcal{L}\left( {y_{1,1}(t)} \right)}} \\{= {{\mathcal{L}\left( {h_{1}(t)} \right)} \cdot {\mathcal{L}\left( \left( {{\overset{\sim}{x}(t)}^{j\; \omega_{c}t}} \right) \right)}}} \\{= {{H_{1}(s)}{{\overset{\sim}{X}\left( {s - {j\; \omega_{c}}} \right)}.}}}\end{matrix}} & (16) \\\begin{matrix}{{Y_{1,3}\left( {s_{1},s_{2},s_{3}} \right)} = {\mathcal{L}\left( {y_{1,3}\left( {t_{1},t_{2},t_{3}} \right)} \right)}} \\{= {{\mathcal{L}\left( {h_{3}\left( {t_{1},t_{2},t_{3}} \right)} \right)} \cdot}} \\{\left\{ {{\mathcal{L}\left( {\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)^{*}} \right)} +} \right.} \\{{{\mathcal{L}\left( {\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)^{*}\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)} \right)} +}} \\\left. {\mathcal{L}\left( {\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)^{*}\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)} \right)} \right\} \\{= {{\mathcal{L}\left( {h_{3}\left( {t_{1},t_{2},t_{3}} \right)} \right)} \cdot}} \\{\left\{ {{{\mathcal{L}\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)}{\mathcal{L}\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)}{\mathcal{L}\left( \left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)^{*} \right)}} +} \right.} \\{{{{\mathcal{L}\left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)}{\mathcal{L}\left( \left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)^{*} \right)}{\mathcal{L}\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)}} +}} \\\left. {{\mathcal{L}\left( \left( {{\overset{\sim}{x}\left( t_{1} \right)}^{j\; \omega_{c}t_{1}}} \right)^{*} \right)}{\mathcal{L}\left( {{\overset{\sim}{x}\left( t_{2} \right)}^{j\; \omega_{c}t_{2}}} \right)}{\mathcal{L}\left( {{\overset{\sim}{x}\left( t_{3} \right)}^{j\; \omega_{c}t_{3}}} \right)}} \right\} \\{{= {H_{3}\left( {s_{1},s_{2},s_{3}} \right)}} \cdot} \\{\left\{ {{{\overset{\sim}{X}\left( {s_{1} - {j\; \omega_{c}}} \right)}{\overset{\sim}{X}\left( {s_{2} - {j\; \omega_{c}}} \right)}{{\overset{\sim}{X}}^{*}\left( \left( {s_{3} - {j\; \omega_{c}}} \right)^{*} \right)}} +} \right.} \\{{{{\overset{\sim}{X}\left( {s_{1} - {j\; \omega_{c}}} \right)}{{\overset{\sim}{X}}^{*}\left( \left( {s_{2} - {j\; \omega_{c}}} \right)^{*} \right)}{\overset{\sim}{X}\left( {s_{3} - {j\; \omega_{c}}} \right)}} +}} \\\left. {{{\overset{\sim}{X}}^{*}\left( \left( {s_{1} - {j\; \omega_{c}}} \right)^{*} \right)}{\overset{\sim}{X}\left( {s_{2} - {j\; \omega_{c}}} \right)}{\overset{\sim}{X}\left( {s_{3} - {j\; \omega_{c}}} \right)}} \right\}\end{matrix} & (17)\end{matrix}$

The above derivation uses the distributive and product transformproperties of the multi-variate Laplace transform. The BBE form ofequations (16) and (17) can be derived by applying a frequencytranslation of jω_(c), to the constituent terms Y_(1,1) and Y_(1,3) asfollows:

$\begin{matrix}{\mspace{79mu} \begin{matrix}{{{\overset{\sim}{Y}}_{1,1}(s)} = {Y_{1,1}\left( {s + {j\; \omega_{c}}} \right)}} \\{= {{H_{1}\left( {s + {j\; \omega_{c}}} \right)}{\overset{\sim}{X}(s)}}} \\{= {{{\overset{\sim}{H}}_{1}(s)}{{\overset{\sim}{X}(s)}.}}}\end{matrix}} & (18) \\\begin{matrix}{{{\overset{\sim}{Y}}_{1,3}\left( {s_{1},s_{2},s_{3}} \right)} = {Y_{1,3}\left( {{s_{1} + {j\; \omega_{c}}},{s_{2} + {j\; \omega_{c}}},{s_{3} + {j\; \omega_{c}}}} \right)}} \\{= {{H_{3}\left( {{s_{1} + {j\; \omega_{c}}},{s_{2} + {j\; \omega_{c}}},{s_{3} + {j\; \omega_{c}}}} \right)} \cdot}} \\{\left\{ {{{\overset{\sim}{X}\left( s_{1} \right)}{\overset{\sim}{X}\left( s_{2} \right)}{{\overset{\sim}{X}}^{*}\left( \left( s_{3} \right)^{*} \right)}} + {{\overset{\sim}{X}\left( s_{1} \right)}{{\overset{\sim}{X}}^{*}\left( \left( s_{2} \right)^{*} \right)}{\overset{\sim}{X}\left( s_{3} \right)}} +} \right.} \\\left. {{{\overset{\sim}{X}}^{*}\left( \left( s_{1} \right)^{*} \right)}{\overset{\sim}{X}\left( s_{3} \right)}{\overset{\sim}{X}\left( s_{2} \right)}} \right\} \\{= {{{\overset{\sim}{H}}_{3}\left( {s_{1},s_{2},s_{3}} \right)} \cdot \left\{ {{{\overset{\sim}{X}\left( s_{1} \right)}{\overset{\sim}{X}\left( s_{2} \right)}{{\overset{\sim}{X}}^{*}\left( \left( s_{3} \right)^{*} \right)}} +} \right.}} \\\left. {{{\overset{\sim}{X}\left( s_{1} \right)}{{\overset{\sim}{X}}^{*}\left( \left( s_{2} \right)^{*} \right)}{\overset{\sim}{X}\left( s_{3} \right)}} + {{{\overset{\sim}{X}}^{*}\left( \left( s_{1} \right)^{*} \right)}{\overset{\sim}{X}\left( s_{3} \right)}{\overset{\sim}{X}\left( s_{2} \right)}}} \right\}\end{matrix} & \;\end{matrix}$

where {tilde over (Y)}_(1,1) and {tilde over (Y)}_(1,3) designate theBBE outputs and {tilde over (H)}₁ and {tilde over (H)}₃ denote the BBEkernels, respectively. Applying the inverse Laplace to equation (18)yields the continuous-time BBE model as follows:

$\begin{matrix}{{{\overset{\sim}{y}}_{1,1}(t)} = {\int_{- \infty}^{\infty}{{{\overset{\sim}{h}}_{1}\left( \tau_{1} \right)} \cdot {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \cdot {{\tau_{1}}.\begin{matrix}{{{\overset{\sim}{y}}_{1,3}(t)} = {\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{\int_{- \infty}^{\infty}{{{\overset{\sim}{h}}_{3}\left( {\tau_{1},\tau_{2},\tau_{3}} \right)} \cdot}}}}} \\{{{\left\{ {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)^{*}} +}} \\{{{\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} +}} \\{{\left. {\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} \right\} \cdot {\tau_{3}}}{\tau_{2}}{{\tau_{1}}.}}\end{matrix}}}}} & (19)\end{matrix}$

Step 6: Discrete-Time Baseband Equivalent Volterra Series Model.

Using the signal and system causality, the integral bounds of the modelof equation (19) are adjusted to ∞→t and −∞→0, respectively. The modelis then given by:

$\begin{matrix}{{{\overset{\sim}{y}}_{1,1}(t)} = {\int_{0}^{t}{{{\overset{\sim}{h}}_{1}\left( \tau_{1} \right)} \cdot {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \cdot {{\tau_{1}}.\begin{matrix}{{{\overset{\sim}{y}}_{1,3}(t)} = {\int_{0}^{t}{\int_{0}^{t}{\int_{0}^{t}{{{\overset{\sim}{h}}_{3}\left( {\tau_{1},\tau_{2},\tau_{3}} \right)} \cdot}}}}} \\{{{\left\{ {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)^{*}} +}} \\{{{\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} +}} \\{{\left. {\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} \right\} \cdot {\tau_{3}}}{\tau_{2}}{\tau_{1}}}\end{matrix}}}}} & (20)\end{matrix}$

Using the fading memory assumption for the steady-state response of thePA (t>T_(∞)) (the transient-response, time-invariant Volterra series isdefined as t<T_(∞)), the corresponding model can be represented as:

$\begin{matrix}{{{\overset{\sim}{y}}_{1,1}(t)} = {\int_{0}^{T_{\infty}}{{{\overset{\sim}{h}}_{1}\left( \tau_{1} \right)} \cdot {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \cdot {{\tau_{1}}.\begin{matrix}{{{\overset{\sim}{y}}_{1,3}(t)} = {\int_{0}^{T_{\infty}}{\int_{0}^{T_{\infty}}{\int_{0}^{T_{\infty}}{{{\overset{\sim}{h}}_{3}\left( {\tau_{1},\tau_{2},\tau_{3}} \right)} \cdot}}}}} \\{{{\left\{ {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)^{*}} +}} \\{{{\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} +}} \\{{\left. {\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} \right\} \cdot {\tau_{3}}}{\tau_{2}}{{\tau_{1}}.}}\end{matrix}}}}} & (21)\end{matrix}$

Using the symmetry of the integrated function (distortion components aresymmetrical and Volterra kernels can be symmetrized), the model can besimplified to:

$\begin{matrix}{{{\overset{\sim}{y}}_{1,1}(t)} = {\int_{0}^{T_{\infty}}{{{\overset{\sim}{h}}_{1}\left( \tau_{1} \right)} \cdot {{\overset{\sim}{x}}_{1}\left( {t - \tau_{1}} \right)} \cdot {{\tau_{1}}.\begin{matrix}{{{\overset{\sim}{y}}_{1,3}(t)} = {\int_{0}^{T_{\infty}}{\int_{\tau_{1}}^{T_{\infty}}{\int_{\tau_{2}}^{T_{\infty}}{{{\overset{\sim}{h}}_{3}\left( {\tau_{1},\tau_{2},\tau_{3}} \right)} \cdot}}}}} \\{{{\left\{ {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)^{*}} +}} \\{{{\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} +}} \\{{\left. {\left( {\overset{\sim}{x}\left( {t - \tau_{1}} \right)} \right)^{*}\left( {\overset{\sim}{x}\left( {t - \tau_{2}} \right)} \right)\left( {\overset{\sim}{x}\left( {t - \tau_{3}} \right)} \right)} \right\} \cdot {\tau_{3}}}{\tau_{2}}{\tau_{1}}}\end{matrix}}}}} & (22)\end{matrix}$

Digitization of the model yields:

$\begin{matrix}{\mspace{79mu} {{{{\overset{\sim}{y}}_{1,1}(n)} = {\sum\limits_{i_{1} = 0}^{M\; 1}{{\overset{\sim}{h}}_{i_{1}}{{\overset{\sim}{x}}_{1}\left( {n,i_{1}} \right)}}}}\mspace{79mu} {{{\overset{\sim}{y}}_{1,3}(n)} = {\sum\limits_{i_{1} = 0}^{M\; 3}{\sum\limits_{i_{2} = i_{1}}^{M\; 3}{\sum\limits_{i_{3} = i_{2}}^{M\; 3}{{\overset{\sim}{h}}_{i_{1},i_{2},i_{3}} \cdot {{\overset{\sim}{x}}_{3}\left( {n,i_{1},i_{2},i_{3}} \right)}}}}}}\mspace{79mu} {{{\overset{\sim}{x}}_{1}\left( {n,i_{1}} \right)} = {\overset{\sim}{x}\left( {n - i_{1}} \right)}}{{{\overset{\sim}{x}}_{3}\left( {n,i_{1},i_{2},i_{3}} \right)} = {{{\overset{\sim}{x}\left( {n - i_{1}} \right)}{\overset{\sim}{x}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{3}} \right)}} + {{\overset{\sim}{x}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}} + {{{\overset{\sim}{x}}^{*}\left( {n - i_{1}} \right)}{\overset{\sim}{x}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}}}}}} & (23)\end{matrix}$

where M₁, M₃ denote the memory depth of the first and third, orderVolterra series. Similarly to the above derivation, it can be shown thatthe fifth order Volterra kernel, {tilde over (y)}_(1,5)(n), is given by:

$\begin{matrix}{{{{\overset{\sim}{y}}_{1,5}(n)} = {\sum\limits_{i_{1} = 0}^{M\; 5}{\sum\limits_{i_{2} = i_{1}}^{M\; 5}{\sum\limits_{i_{3} = i_{2}}^{M\; 5}{\sum\limits_{i_{4} = i_{3}}^{M\; 5}{\sum\limits_{i_{5} = i_{4}}^{M\; 5}{{\overset{\sim}{h}}_{i_{1},i_{2},i_{3},i_{4},i_{5}} \cdot {{\overset{\sim}{x}}_{5}\left( {n,i_{1},i_{2},i_{3},i_{4},i_{5}} \right)}}}}}}}}{{\overset{\sim}{x}}_{5}\left( {n,i_{1},i_{2},i_{3},i_{4},i_{5}} \right)} = {{{\overset{\sim}{x}\left( {n - i_{1}} \right)}{\overset{\sim}{x}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{5}} \right)}} + {{\overset{\sim}{x}\left( {n - i_{1}} \right)}{\overset{\sim}{x}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{3}} \right)}{\overset{\sim}{x}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{5}} \right)}} + {{\overset{\sim}{x}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}{\overset{\sim}{x}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}^{*}\left( {n - i_{1}} \right)}{\overset{\sim}{x}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}{\overset{\sim}{x}\left( {n - i_{4}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{5}} \right)}} + {{\overset{\sim}{x}\left( {n - i_{1}} \right)}{\overset{\sim}{x}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{4}} \right)}{\overset{\sim}{x}\left( {n - i_{5}} \right)}} + {{\overset{\sim}{x}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{4}} \right)}{\overset{\sim}{x}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{4}} \right)}{\overset{\sim}{x}\left( {n - i_{5}} \right)}} + {{\overset{\sim}{x}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{3}} \right)}{\overset{\sim}{x}\left( {n - i_{4}} \right)}{\overset{\sim}{x}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{2}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{3}} \right)}{\overset{\sim}{x}\left( {n - i_{4}} \right)}{\overset{\sim}{x}\left( {n - i_{5}} \right)}} + {{{\overset{\sim}{x}}^{*}\left( {n - i_{1}} \right)}{{\overset{\sim}{x}}^{*}\left( {n - i_{2}} \right)}{\overset{\sim}{x}\left( {n - i_{3}} \right)}{\overset{\sim}{x}\left( {n - i_{4}} \right)}{\overset{\sim}{x}\left( {n - i_{5}} \right)}}}} & (24)\end{matrix}$

where M₅ denotes the memory depth of the fifth order Volterra series.The complex valued BBE Volterra series in equations (23) and (24) up tononlinear distortion products 5 is then given by:

{tilde over (y)} ₁(n)={tilde over (y)} _(1,1)(n)+{tilde over (y)}_(1,3)(n)+{tilde over (y)} _(1,5)(n)  (25)

Higher order distortion products can be similarly derived. In addition,only odd powered terms are retained. The terms {tilde over (x)}₃(n, i₁,i₂, i₃) and {tilde over (x)}₅(n, i₁, i₂, i₃, i₄, i₅) are linearcombinations of 3 third, and 10 fifth, order distortion products. Anexample of the BBE Volterra model of equation (25) is given for NL=3 andM=1 as follows

$\begin{matrix}{{\overset{\sim}{y}(n)} = {{\frac{1}{2}h_{0}{\overset{\sim}{x}(n)}} + {\frac{1}{2}h_{1}{\overset{\sim}{x}\left( {n - 1} \right)}} + {\frac{3}{8}h_{0,0,0}{\overset{\sim}{x}(n)}{\overset{\sim}{x}(n)}{{\overset{\sim}{x}}^{*}(n)}} + {\frac{1}{8}{h_{0,0,1}\left( {{2{\overset{\sim}{x}(n)}{\overset{\sim}{x}\left( {n - 1} \right)}{{\overset{\sim}{x}}^{*}(n)}} + {{\overset{\sim}{x}(n)}{\overset{\sim}{x}(n)}{{\overset{\sim}{x}}^{*}\left( {n - 1} \right)}}} \right)}} + {\frac{1}{8}{h_{0,1,1}\left( {{2{\overset{\sim}{x}(n)}{\overset{\sim}{x}\left( {n - 1} \right)}{{\overset{\sim}{x}}^{*}\left( {n - 1} \right)}} + {{\overset{\sim}{x}\left( {n - 1} \right)}{\overset{\sim}{x}\left( {n - 1} \right)}{{\overset{\sim}{x}}^{*}(n)}}} \right)}} + {\frac{3}{8}h_{1,1,1}{\overset{\sim}{x}\left( {n - 1} \right)}{\overset{\sim}{x}\left( {n - 1} \right)}{{\overset{\sim}{x}}^{*}\left( {n - 1} \right)}}}} & (26)\end{matrix}$

Comparison of the classical expression of equation (5) with the newlyproposed BBE Volterra expression of equation (26) described hereinresults in several conclusions. For example, the proposed BBE Volterramodel effectively combines a number of distortion components to share aunique kernel, as per equations (23) and (24). For example, the twodistortion components {tilde over (x)}(n){tilde over (x)}(n−1){tildeover (x)}*(n−1) and {tilde over (x)}(n−1){tilde over (x)}(n−1){tildeover (x)}*(n) share the same kernel {tilde over (h)}_(0,1,1) in the BBEVolterra model of equation (26), however, they have different kernels({tilde over (h)}_(0,1,1) and {tilde over (h)}_(1,1,0), respectively) inthe LPE Volterra model of equation (5). More generally, severaldistortion products, which were assigned distinct kernels in equation(4), are linearly combined to form a novel term in the new model spaceproposed by equation (25), and hence, share the same kernel.

Table I summarizes the number of kernels needed for four differentexample modeling scenarios where the nonlinearity order has been set toseven and the memory depth M, equals M1=M3=M5=M. The expressions for thenumber of coefficients were used to plot FIG. 2. The plot shows thenumber of kernels for different memory depths for DDE LPE Volterra r=210, r=3 12, compact BBE Volterra 14 and Classical LPE Volterra 16.

As can be seen, the newly proposed, compact model involved asignificantly lower number of coefficients than the full classical LPEVolterra series, especially as the memory depth increased. This resultsin lower complexity and cost of implementation. However, the pruned LPEVolterra using the DDR approach also required a smaller number ofcoefficients as it discarded distortion products with a dynamicorder≧three.

TABLE 1 Number of coefficients for NL = 7 Classical LPE Volterra$4 + {\frac{43}{4}M} + {\frac{451}{36}M^{2}} + {\frac{1213}{144}M^{3}} + {\frac{121}{36}M^{4}} + {\frac{59}{72}M^{5}} + {\frac{1}{9}M^{6}} + {\frac{1}{144}M^{7}}$BBE Volterra$4 + {\frac{1619}{210}M} + {\frac{1973}{360}M^{2}} + {\frac{1597}{720}M^{3}} + {\frac{37}{72}M^{4}} + {\frac{13}{180}M^{5}} + {\frac{1}{180}M^{6}} + {\frac{1}{5040}M^{7}}$DDR LPE Volterra R = 2 $4 + {\frac{25}{2}M} + {\frac{11}{2}M^{2}}$ DDRLPE Volterra R = 3$4 + {\frac{131}{6}M} + {\frac{39}{2}M^{2}} + {\frac{14}{3}M^{3}}$

Thus, different orders of the nonlinear dynamic response, {tilde over(y)}_(1,1)(n), {tilde over (y)}_(1,3)(n) and {tilde over (y)}_(1,5)(n)are results of different nonlinear integrals {tilde over (y)}_(1,1)(t),{tilde over (y)}_(1,3)(t) and {tilde over (y)}_(1,5)(t). Depending onthe dynamic property of the PA, and depending on the numericalintegration scheme used, different integrals may call for differentmemory depths. Hence, in the proposed BBE Volterra model of equation(25), three memory depth parameters may be defined, M1, M3 and M5,instead of the single memory depth parameter, M, used in the LPEVolterra model of equation (4). This allows a flexible control of themodel's dynamics based on a theoretical framework. Note also that, dueto the symmetry properties of the integrals in each order of thenonlinear dynamic response, the different summation operators in {tildeover (y)}_(1,1)(n), {tilde over (y)}_(1,3)(n) and {tilde over(y)}_(1,5)(n) use the same memory depth M1, M3 and M5 respectively.

The signal transformations used to derive the BBE expression of equation(25) preserve the linear property of the Volterra series with respect toits coefficients. Therefore, commonly used model identificationalgorithms such as the least square error (LSE) can be applied toidentify the kernels in equation (25) for a given RF PA. Equation (27)suggests the expression used to deduce the LSE solution of equation (25)

X·h=Y  (27)

where X, h and Y denote the input signal non-square matrix, the vectorthat contains the unknown kernels and the output signal vector asdescribed below:

$\begin{matrix}{{X = \begin{pmatrix}{{\overset{\sim}{x}}_{0}(1)} & \ldots & {{\overset{\sim}{x}}_{M}(1)} & {{\overset{\sim}{x}}_{0,0,0}(1)} & \ldots & {{\overset{\sim}{x}}_{M,M,M}(1)} & \; \\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots & \ldots \\{{\overset{\sim}{x}}_{0}(N)} & \ldots & {{\overset{\sim}{x}}_{M}(N)} & {{\overset{\sim}{x}}_{0,0,0}(N)} & \ldots & {{\overset{\sim}{x}}_{M,M,M}(N)} & \;\end{pmatrix}}{{{h = \begin{pmatrix}h_{0} \\\vdots \\h_{M} \\\vdots\end{pmatrix}};{Y = \begin{pmatrix}{\overset{\sim}{y}(1)} \\\vdots \\{\overset{\sim}{y}(N)}\end{pmatrix}}},}} & (28)\end{matrix}$

In the above expression, N represents the total number of measurementdata. The solution to the problem defined in equation (27) is:

{circumflex over (h)}=(X ^(t) ·X)⁻¹ ·X ^(t) ·Y  (29)

where ĥ is the estimate of h.

To summarize the approach, a discrete BBE Volterra series is generatedbased on a received wideband or intra-band carrier aggregated signal.The series has distortion products grouped according to determinedshared kernels. The shared kernels are determined based on atransformation of a real-valued continuous-time pass band Volterraseries without pruning of kernels. The transformation includestransforming the real-valued continuous time pass band Volterra seriesto a multi-frequency complex-valued envelope series. The multi-frequencycomplex-valued envelope signal is then transformed to a continuous-timepass band-only series. The continuous-time pass band-only signal istransformed to a continuous-time baseband equivalent series. Thecontinuous-time baseband equivalent signal is discretized to produce thediscrete base band equivalent Volterra series. Shared kernels of thediscrete base band equivalent Volterra series are identified, where eachshared kernel has distortion products in common with another sharedkernel.

To assess the performance of the newly proposed compact BBE Volterramodel, the model was used to linearize different PAs driven withwideband and intra-band carrier aggregated signals. The linearizationperformance was compared with that of the classical LPE Volterra modeland the DDR pruned Volterra model.

By way of example, for the wideband signal linearization test, a 200 WDoherty PA (DPA) driven with a 2C 1001 WCDMA signal was linearized. Thedevice under test (DUT) was excited with a 1 ms signal sampled at 100MHz, where, out of 100 k points, 10 k were used to train the proposedmodel. Being non-pruned Volterra models, the nonlinearity order andmemory depth for both the proposed model and the LPE Volterra model wereset to be equal. Hence, the memory depths of the different nonlineardynamic responses of the BBE Volterra model were set at the sameM1=M3=M5=M. However, the DDR pruned Volterra model parameters werevaried separately for optimum performance. The different models'parameters and the resulting model complexities are summarized in TableII. It can be seen that both the DDR pruned Volterra and the BBEVolterra models have a significantly lower number of kernels than theclassical LPE Volterra model. Table II also summarizes the linearizationresults of the DUT.

TABLE II COMPLEXITY AND LINEARIZATION PERFORMANCE OF A DOHERTY POWERAMPLIFIER USING DIFFERENT VOLTERRA FORMULATIONS Number of ModelParameters Coefficients EVM ACLR (dBc) Without  6% −38 DPD Classical NL= 7, M = 2 231 1.2% −55 LPE Volterra DDR LPE NL = 7, M = 3, 91 1.3% −54Volterra r = 2 BBE NL = 7, M = 2 70 1.2% −55 Volterra

The spectrum plot of the PA response without predistortion, and underthe linearization of the different Volterra formulations, is presentedin FIG. 3. It shows that all of the tested models successfully reducedthe error vector magnitude (EVM) from 6% to about 1.2% and the adjacentchannel leakage ratio (ACLR) from −38 dBc to about −55 dBc. It can beconcluded that both the DDR pruning of the LPE Volterra model and thekernel sharing of the BBE Volterra model did not affect thelinearization performance of the classical LPE Volterra formulation.

In order to validate the proposed model under recent 4G communicationsignals, an intra-band carrier aggregated signal was synthesized. Thissignal was mixed-standard, with a bandwidth of 40 MHz and a PAPR of 9.8dB, containing a 15 MHz Long Term Evolution (LTE) signal and a 15 MHz101 Wideband Code Division Multiple Access (WCDMA) signal separated by a10 MHz guard band. Two different RF PA demonstrators were used as DUT.The first was a single ended Class J 45 W GaN amplifier and the secondwas a laterally diffused MOSFET (LDMOS) 200 W Doherty PA. The DUT wasexcited with a 1 ms signal sampled at 122.88 MHz, where, out of 122.88 kpoints, 10 k were used to train the proposed model.

Tables III and IV summarize the linearization results found with eachamplifier using both the proposed compact and the pruned LPE Volterramodels. For all testing, the nonlinearity order was set to seven and thememory depth was set two and three.

TABLE III COMPLEXITY AND LINEARIZATION PERFORMANCE OF A CLASS AB GANPOWER AMPLIFIER USING DIFFERENT VOLTERRA FORMULATIONS Number of ModelParameters Coefficients EVM ACLR (dBc) Without 5.5% −35 DPD Classical NL= 7, M = 2 231  2% −48 LPE Volterra DDR LPE NL = 7, M = 3, 91 2.1% −48Volterra r = 2 BBE NL = 7, M = 2 70  2% −48 Volterra

TABLE IV COMPLEXITY AND LINEARIZATION PERFORMANCE OF LDMOS DOHERTY POWERAMPLIFIER USING DIFFERENT VOLTERRA FORMULATIONS Number of ModelParameters Coefficients EVM ACLR (dBc) Without 5.5% −35 DPD Classical NL= 7, M = 2 231 1.9% −50 LPE Volterra DDR LPE NL = 7, M = 3, 91 2.3% −46Volterra r = 2 BBE NL = 7, M = 2 70 1.9% −50 Volterra

According to Table III, both the pruned LPE Volterra and the compactmodels allowed for similar linearization performance as the EVM wasreduced from 5.5% to 2% and the ACLR was increased from 35 dB to about50 dB (after application of the linearization). This linearizationcapacity was successfully achieved despite the significant reduction inthe number of kernels required by the compact model (91 in the prunedLPE Volterra model versus 70 in the proposed model).

It is worth mentioning that the full LPE Volterra model required 231kernels. The linearization performance was further confirmed byexamining the spectrum of the 45 W GaN PA output signals with andwithout predistortion. This is shown in FIG. 4, which is the spectrum ofthe single-ended GaN PA driven with the 40 MHz multi-standard intra-bandcarrier aggregated signal using the different Volterra seriesformulations and without linearization.

In the case of the second RF PA (see Table IV), the pruned LPE Volterraand the proposed model showed different linearization performance. Whileboth the classical LPE Volterra model and the compact BBE Volterra modelachieved similar linearization results in terms of EVM (1.9%) and ACLR(−50 dBc), the DDR LPE Volterra led to a 0.4% higher EVM and 4 dB lowerACLR. The superiority of the linearization capacity of the new compactmodel is confirmed in FIG. 5, which displays the spectra of the LDMOSDoherty PA excited by the 40 MHz multi-standard intra-band carrieraggregated signal using different Volterra series formulations andwithout linearization.

A noticeable residual out of band spectrum regrowth in the signalobtained using the pruned DDR LPE can be observed. As previouslymentioned, the DDR pruning approach may suffer from reducedlinearization capability when the dynamic order is set to two as iteliminates a large number of distortion products. The compact BBEVolterra model described herein does not suffer from this limitation.All distortion products are maintained. The pruned LPE Volterralinearization performance could be further improved if the dynamic orderR were increased further, however, this would result in a larger numberof kernels, e.g., the number of kernels would be 162 for r=3 M=2.

FIG. 6 is a block diagram of a power amplification system 18 having adigital pre-distorter modelling unit 34 implementing the BBE Volterramodel presented herein. The power amplification system 18 includes adigital pre-distorter 20. The DPD 20 receives the identified sharedkernels from the pre-distorter modelling unit 34, and pre-distorts theinput signal xto produce a pre-distorted signal. The pre-distortedsignal is converted to analog by a digital to analog (DAC) converter 22,low pass filtered by a filter 24, and mixed to radio frequency (RF) by amixer 26 to prepare the signal for amplification by an RF PA amplifier28. A directional coupler 30 couples the output of the RF PA 28 to atransmitter observation receiver (TOR) 32 The TOR 32 samples the outputof the power amplifier 28 in each band and produces a TOR output signal,y. The TOR output signal is used by the DPD modeling unit 34 to derivethe kernel vector h according to equation (27). The DPD modelling unit34 calculates a discrete baseband equivalent Volterra series havingdistortion products grouped according to determined shared kernels,where the shared kernels are based on a transformation of a real-valuedcontinuous-time pass band Volterra series without pruning of kernels.

FIG. 7 is a detailed view of the DPD modelling unit 34, which includes aprocessor 36 in communication with a memory module 38. The processor 36may be implemented as a micro-processor operating according to computerinstructions organized as a group of software modules, or each module ofthe processor 36 may be implemented by application specific integratedcircuitry. The memory 38 may be implemented as random access memory(RAM), for example, and may contain non-volatile components, such asread only memory (ROM) that stores programmatic code to implement thefunctions described herein. The DPD modelling unit 34 receives the inputsignal xand the TOR output signal yfrom the transmitter observationreceiver 32 and derives the modelling vector h according to equation(27). The modelling vector his input to the DPD 20 to pre-distort theinput signal xto produce a pre-distorted signal. The processor 36includes a grouping module 40, a shared kernel determiner 42 and aseries term computer 44. The grouping module 40 is configured to groupdistortion products of the series according to determined sharedkernels, as shown in equation (25). The shared kernel determiner 42 isconfigured to determine the shared kernels based on a transformation ofa real-valued continuous-time pass band Volterra series without pruningof kernels, as described above with reference to Step 6. The series termcalculator 44 is configured to calculate the terms of the discrete baseband equivalent Volterra series according to equation (27), the termsbeing the distortion products multiplied by their respective sharedkernels. The memory module 38 is configured to store terms of thediscrete BBE Volterra series 46, generated by the processor 36.

FIG. 8 is a flowchart of an exemplary process for modelling a poweramplifier 40 fed by a wideband or intra-band carrier aggregated inputsignal using the discrete BBE Volterra series described herein forimproved linearization with reduced complexity. The input signal isreceived by a digital pre-distorter 28 (block S100). A discrete BBEVolterra series is generated by the DPD modelling unit 34 based on thereceived input signal (block S102), according to equations (25) and(27). The series has distortion products that are grouped by thegrouping module 40 according to determined shared kernels determined byshared kernel determiner 42, as shown in equation (25). The sharedkernels are determined based on a transformation of a real-valuedcontinuous-time pass band Volterra series without pruning of kernels,according to Steps 1 through 6 above.

FIG. 9 is a flowchart of an exemplary process for transforming areal-valued continuous time pass band Volterra series to a discrete baseband equivalent Volterra series according to Steps 1 through 6 above inwhich shared kernels are identified as set out in block S102 of FIG. 8.The real-valued continuous time pass band Volterra series is transformedto a multi-frequency complex-valued envelope series, as in Steps 1 and2, described above (block S104). The multi-frequency complex-valuedenvelope signal is transformed to a continuous-time pass band-onlyseries, as in Step 3 described above (block S106). The continuous-timepass band-only signal is transformed to a continuous-time basebandequivalent series, as in Steps 4 and 5 described above (block S108). Thecontinuous-time baseband equivalent signal is discretized to produce thediscrete base band equivalent Volterra series, as in Step 6 describedabove. (block S110). Shared kernels of the discrete base band equivalentVolterra series are identified, where a shared kernel has distortionproducts in common with another shared kernel, also described as in Step6 described above (block S112).

FIG. 10 is a flowchart of an exemplary process of transforming thecontinuous-time pass band-only signal to a continuous-time basebandequivalent signal as shown in block S108 of FIG. 9, as in Step 5described above. The continuous-time pass band-only series is expressedin convolution form (block S114). Then, the Laplace transform is appliedto the convolution form to produce a Laplace domain expression (blockS116). A number of terms in the Laplace domain expression may be reducedbased on symmetry (block S118). The Laplace domain expression isfrequency-shifted to baseband to produce a baseband equivalentexpression in the Laplace domain (block S120). An inverse Laplacetransform is applied to the baseband equivalent expression to producethe continuous-time baseband equivalent series (block S122).

FIG. 11 is a flowchart of a process of discretizing the continuous-timebaseband equivalent series to produce the discrete base band equivalentVolterra series, as shown in block S110 of FIG. 9, as in Step 6described above. The process includes truncating the continuous-timebaseband equivalent series to a finite non-linearity order (block S124).The process also includes expressing the truncated series as summationsof non-linear distortion terms, with upper limits of the summationsbeing memory depths assigned to each order of the non-linear distortionterms (block S126).

The PA modelling method described herein using the discrete BBE Volterraseries is inherently compact and calls for significantly fewercoefficients then its LPE counterpart and therefore avoids theaccuracy-compromising pruning transformations widely applied toclassical formulations in the literature. The validation describedherein proved the excellent modeling and linearization performance ofthe method when compared to the classical non-pruned Volterra model. Thenew formulation also outperformed pruned Volterra models while using alower number of coefficients.

It will be appreciated by persons skilled in the art that the presentinvention is not limited to what has been particularly shown anddescribed herein above. In addition, unless mention was made above tothe contrary, it should be noted that all of the accompanying drawingsare not to scale. A variety of modifications and variations are possiblein light of the above teachings without departing from the scope of thefollowing claims.

1. A method of modelling a power amplifier fed by one of a widebandsignal and an intra-band carrier aggregated signal, the methodcomprising: receiving the one of the wideband signal and the intra-bandcarrier aggregated signal; generating a discrete base band equivalent,BBE, Volterra series based on the received signal, the series havingdistortion products grouped according to determined shared kernels; andthe shared kernels being determined based on a transformation of areal-valued continuous-time pass band Volterra series without pruning ofkernels.
 2. The method of claim 1, wherein the shared kernels aredetermined based on the transformation of the real-valuedcontinuous-time pass band Volterra series by: transforming thereal-valued continuous time pass band Volterra series to amulti-frequency complex-valued envelope series; transforming themulti-frequency complex-valued envelope signal to a continuous-time passband-only series; transforming the continuous-time pass band-only signalto a continuous-time baseband equivalent series; discretizing thecontinuous-time baseband equivalent signal to produce the discrete baseband equivalent Volterra series; and identifying the shared kernels,each shared kernel having distortion products in common with anothershared kernel.
 3. The method of claim 2, wherein transforming thecontinuous-time pass band-only signal to a continuous-time basebandequivalent signal includes: expressing the continuous-time passband-only series in convolution form; applying a Laplace transform tothe convolution form to produce a Laplace domain expression; frequencyshifting the Laplace domain expression to baseband to produce a basebandequivalent expression in the Laplace domain; and applying an inverseLaplace transform to the baseband equivalent expression to produce thecontinuous-time baseband equivalent series.
 4. The method of claim 3,wherein a number of terms in the Laplace domain expression are reducedvia symmetry.
 5. The method of claim 3, further comprising groupingterms of the Laplace domain expression based on frequency intervalswhere distortion terms are not zero.
 6. The method of claim 2, whereindiscretizing the continuous-time baseband equivalent series to producethe discrete base band equivalent Volterra series includes: truncatingthe continuous-time baseband equivalent series to a finite non-linearityorder; and expressing the truncated series as summations of non-lineardistortion terms, with upper limits of the summations being memorydepths, different memory depths being assignable to different ones ofthe summations.
 7. The method of claim 6, wherein a distortion term is agroup of distortion products multiplied by a shared kernel.
 8. A digitalpre-distorter (DPD) system for pre-distortion of one of a widebandsignal and an intra-band carrier aggregated signal, comprising: aVolterra series DPD modelling unit, the DPD modelling unit configuredto: receive the one of the wideband signal and the intra-band carrieraggregated signal; and calculate a discrete base band equivalent, BBE,Volterra series based on the received signal, the series havingdistortion products grouped according to determined shared kernels; andthe shared kernels being determined based on a transformation of areal-valued continuous-time pass band Volterra series without pruning ofkernels.
 9. The DPD system of claim 8, further comprising: a poweramplifier, the power amplifier configured to produce an output inresponse to the one of the wideband signal and the intra-band carrieraggregated received signal, the output of the power amplifier providedto the Volterra series DPD modelling unit, the Volterra series DPDmodeling unit being configured to compute the shared kernels based onthe output of the power amplifier.
 10. The DPD system of claim 9,further comprising a transmitter observation receiver configured tosample the output of the power amplifier and provide the sampled outputto the Volterra series DPD modelling unit.
 11. The DPD system of claim8, wherein the distortion products and their associated kernels aredetermined by: transforming the real-valued continuous time pass bandVolterra series to a multi-frequency complex-valued envelope series;transforming the multi-frequency complex-valued envelope signal to acontinuous-time pass band-only series; transforming the continuous-timepass band-only signal to a continuous-time baseband equivalent series;discretizing the continuous-time baseband equivalent signal to producethe discrete base band equivalent Volterra series; and identifying theshared kernels, each shared kernel having distortion products in commonwith another shared kernel.
 12. The DPD system of claim 11, whereintransforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal includes: expressing thecontinuous-time pass band-only series in convolution form; applying aLaplace transform to the convolution form to produce a Laplace domainexpression; frequency shifting the Laplace domain expression to basebandto produce a baseband equivalent expression in the Laplace domain; andapplying an inverse Laplace transform to the baseband equivalentexpression to produce the continuous-time baseband equivalent series.13. The DPD system of claim 12, wherein a number of terms in the Laplacedomain expression are reduced via symmetry.
 14. The DPD system of claim12, wherein transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal further comprises groupingterms of the Laplace domain expression based on frequency intervalswhere distortion terms are not zero.
 15. The DPD system of claim 11,wherein discretizing the continuous-time baseband equivalent series toproduce the discrete base band equivalent Volterra series includes:truncating the continuous-time baseband equivalent series to a finitenon-linearity order; and expressing the truncated series as summationsof non-linear distortion terms, with upper limits of the summationsbeing memory depths assigned to each order of the non-linear distortionterms.
 16. The DPD system of claim 15, wherein a distortion term is agroup of distortion products multiplied by a shared kernel.
 17. AVolterra series digital pre-distorter, DPD, modelling unit, comprising:a memory module, the memory module configured to store terms of adiscrete base band equivalent, BBE, Volterra series, the series beingbased on one of a received wideband signal and an intra-band carrieraggregated signal; a grouping module, the grouping module configured togroup distortion products of the series according to determined sharedkernels; a shared kernel determiner, the shared kernel determinerconfigured to determine the shared kernels based on a transformation ofa real-valued continuous-time pass band Volterra series without pruningof kernels; and a series term calculator, the series term calculatorconfigured to calculate the terms of the discrete base band equivalentVolterra series, the terms being the distortion products multiplied bytheir respective shared kernels.
 18. The Volterra series DPD modellingunit of claim 17, wherein the shared kernel determiner is furtherconfigured to determine the shared kernels via a least squares estimatebased on the received one of the wideband signal and the intra-bandcarrier aggregated signal, and an output of a power amplifier.
 19. TheVolterra series DPD modelling unit of claim 17, wherein the kernels anddistortion products are derived from the real-valued continuous-timepass band Volterra series by: transforming the real-valued continuoustime pass band Volterra series to a multi-frequency complex-valuedenvelope series; transforming the multi-frequency complex-valuedenvelope signal to a continuous-time pass band-only series; transformingthe continuous-time pass band-only signal to a continuous-time basebandequivalent series; discretizing the continuous-time baseband equivalentsignal to produce the discrete base band equivalent Volterra series; andidentifying the shared kernels, each shared kernel having distortionproducts in common.
 20. The Volterra series DPD modelling unit of claim19, wherein transforming the continuous-time pass band-only signal to acontinuous-time baseband equivalent signal includes: expressing thecontinuous-time pass band-only series in convolution form; applying aLaplace transform to the convolution form to produce a Laplace domainexpression; frequency shifting the Laplace domain expression to basebandto produce a baseband equivalent expression in the Laplace domain; andapplying an inverse Laplace transform to the baseband equivalentexpression to produce the continuous-time baseband equivalent series.